The principle here is pretty simple. If you fire a bullet forward, the recoil pushes you back. So if you fire downward, the recoil should push you up.

But can it be done? The short answer is, amazingly: yes. The longer answer is of course a little more involved, but amazingly interesting, combining guns with rocket science. For instance:

[O]ur optimal craft comprises a large number of AK-47s (a minimum of 25 but ideally at least 300) carrying 250 rounds of ammunition each. The largest versions of this craft could accelerate upward to vertical speeds approaching 100 meters per second, climbing over half a kilometer into the air.